System reliability evaluation method for routing policy

ABSTRACT

A system reliability evaluation method for routing policy is disclosed. The single minimal path of the routing policy includes plural arcs between a start node and a terminal node in a flow network. The method includes the steps of providing a virtual network in a computer for simulating the flow network; inputting a transmission requirement, a budget restriction and a time restriction; distributing the transmission requirement in a first minimal path of the virtual network for getting a first feasible probability; if the first minimal path is inactive, distributing the transmission requirement in a second minimal path of the virtual network and getting a first inactive probability of the first minimal path; getting a second feasible probability of the second minimal path; and computing a system feasible probability of the virtual network by the first feasible probability, the first inactive probability and the second feasible probability, defining the system reliability.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The invention relates to a system reliability evaluation method forrouting policy, and especially relates to a system reliabilityevaluation method for routing policy of a stochastic-flow network.

(2) Description of the Prior Art

With diversification of commodities and information services, thefactors of flow, time and cost are progressively valued in networkanalysis, but said factors are set as deterministic. Thus, in operationsresearch, computer science, networking and other areas, the quickestpath problem has attracted great attention of researchers, and saidproblem is for finding a quickest path with minimum transmission time tosend a given amount of data from the unique source to the unique sink.Since then, several variants of quickest path problems are proposed:constrained quickest path problem, the first k quickest paths problem,and all-pairs quickest path problem.

However, two factors, flow and time, are separately dealt inconventional network analysis. For instance, the largest capacity pathproblem deals with flow, and the shortest path problem is for findingthe shortest transmission time if time is regarded as parameter. Twofactors, flow and time, are first combined in the quickest path problem,and the transmission time reduced is an important issue through thereal-life network, especially through computer and telecommunicationnetworks.

Due to the factors of failure, maintenance, occupation, etc, thecapacity of each arc is stochastic in many real flow networks such ascomputer system, telecommunication system, logistic system andtransportation system, etc. Because the capacity is influenced with saidfactors, the capacity should be not fixed. Such a network is named astochastic-flow network. For example, in the computer system, eachcomputer (or switch) represents a node, and each transmission linerepresents an arc. The transmission line is composed by plural real-lifenetwork lines (such as coaxial cables, fiber optics, etc), and eachnetwork line has two cases of normal and failure; that implies thetransmission line has several states in which state k means k physicallines are successful. Thus each transmission line has plural states, andcapacity of each arc has plural values accordingly

Thus, in computer and telecommunication networks, when the given amountof data is sent through several minimal paths, how to find the quickesttransmission method in several various factors, calculate a systemreliability denoted successful probability of transmitting the givenamount of data in time restriction, and accord with the reserved pathsfor contribution to the system reliability. Namely, finding the bestrouting policy in time and cost restriction is an important issue toconduct the system reliability evaluation method for a stochastic-flownetwork.

SUMMARY OF THE INVENTION

Accordingly, the object of the invention is to provide a systemreliability evaluation method for routing policy. With setting therestriction of the transmission time and the transmission cost between astart node and a terminal node in a flow network, calculating theprobability satisfied by the restriction to evaluate the quality ofservice for customer.

In one aspect, the invention provides a system reliability evaluationmethod for routing policy is disclosed. The single minimal path of therouting policy includes plural arcs between a start node and a terminalnode in a flow network, wherein each of the minimal paths is an orderedsequence of the arcs between the start node to the terminal node withoutloops. The method includes the steps of providing a virtual network in acomputer for simulating the flow network; inputting a transmissionrequirement, a budget restriction and a time restriction; distributingthe transmission requirement in a first minimal path of the virtualnetwork and calculating a first feasible probability therein accordingto the transmission requirement and the time restriction; if the firstminimal path is in an unfeasible state, distributing the transmissionrequirement in a second minimal path of the virtual network, andcalculating a first unfeasible probability of the first minimal path;then calculating a second feasible probability of the second minimalpath; by the operation unit, integrating the first feasible probability,the first unfeasible probability and the second feasible probabilityinto a system feasible probability of the virtue network, defined as asystem reliability.

In an experiment, the step further includes: if the first minimal pathis in a feasible state, defining the first feasible probability as thesystem reliability.

In an experiment, the steps of distributing the transmission requirementin one of the minimal paths include: selecting the minimal paths of thevirtual network; calculating a flow in each of the minimal paths; andtransferring the flow in each of the minimal paths into a capacity ofeach of the arcs. Wherein the steps of calculating the flow of theminimal path include: providing a lead time for each of the arcs in theminimal path; with a transmission time lower than or equal to the timerestriction, calculating the flow of the minimal path, wherein thetransmission time equals to the sum of the lead time and thetransmission requirement divided by the flow of the minimal path; andjudging if a lower boundary vector exists, when the flow of the minimalpath is smaller than a maximum capacity of the minimal path.

In an experiment, the steps of calculating the feasible probabilitiesinclude: according to the transmission requirement, the timerestriction, the budget restriction and a transmission time which issmaller than or equal to the time restriction, calculating, a maximumflow in each of the arcs in the minimal path per unit time accordingly,regarded as a capacity of the arc; defining a capacity vector composedby the capacities of the arcs, the capacities being stochastic tocorrespond with the flow distribution state of the flow network;executing a budget check by the operation unit to check if atransmission cost satisfies the budget restriction for sending thetransmission requirement in the minimal path; defining the capacityvector as a lower boundary vector when the transmission cost and thetransmission time of the minimal path are less than or equal to thebudget restriction and the time restriction respectively; andcalculating the probability that the capacity vector of any minimal pathis larger than or equal to the lower boundary vector of the minimalpath, and defining said probability as the feasible probability of theflow network.

Said methods of calculating the feasible probabilities includeinclusion-exclusion rule, disjoint-event method and state-spacedecomposition are applied in.

In an experiment, above system reliability evaluation method furtherincludes: defining an expectation of the transmission ability of theflow network as product of the system reliability and the transmissionrequirement, and defining an expectation of the transmission time of theflow network as product of the system reliability and the timerestriction.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic view of an embodiment of the virtual networkaccording to the present invention.

FIG. 2 is a block diagram of the hardware executing an embodiment of thesystem reliability evaluation method for routing policy according to thepresent invention.

FIG. 3 is a flow chart of the software executing an embodiment of thesystem reliability evaluation method routing policy according to thepresent invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following detailed description of the preferred embodiments,reference is made to the accompanying drawings which form a part hereof,and in which is shown by way of illustration specific embodiments inwhich the invention may be practiced. In this regard, directionalterminology, such as “top,” “bottom,” “front,” “back,” etc., is usedwith reference to the orientation of the Figure(s) being described. Thecomponents of the present invention can be positioned in a number ofdifferent orientations. As such, the directional terminology is used forpurposes of illustration and is in no way limiting. On the other hand,the drawings are only schematic and the sizes of components may beexaggerated for clarity. It is to be understood that other embodimentsmay be utilized and structural changes may be made without departingfrom the scope of the present invention. Also, it is to be understoodthat the phraseology and terminology used herein are for the purpose ofdescription and should not be regarded as limiting. The use of“including,” “comprising,” or “having” and variations thereof herein ismeant to encompass the items listed thereafter and equivalents thereofas well as additional items. Unless limited otherwise, the terms“connected,” “coupled,” and “mounted” and variations thereof herein areused broadly and encompass direct and indirect connections, couplings,and mountings. Similarly, the terms “facing,” “faces” and variationsthereof herein are used broadly and encompass direct and indirectfacing, and “adjacent to” and variations thereof herein are used broadlyand encompass directly and indirectly “adjacent to”. Therefore, thedescription of “A” component facing “B” component herein may contain thesituations that “A” component facing “B” component directly or one ormore additional components is between “A” component and “B” component.Also, the description of “A” component “adjacent to” “B” componentherein may contain the situations that “A” component is directly“adjacent to” “B” component or one or more additional components isbetween “A” component and “B” component. Accordingly, the drawings anddescriptions will be regarded as illustrative in nature and not asrestrictive.

Refer to FIG. 1 for a stochastic-flow network with a start node s and aterminal node t, where N stands for all nodes, a_(i) for all arcs, eacharc a_(i) connecting two nodes in N. The flow network can be aninformation network, a telecommunication network, a logistic network ora transportation network.

The present invention provides a system reliability evaluation methodfor routing policy. The system reliability means the probability thatthe stochastic-flow network is able to send a specific amount of datafrom a single start node to a single terminal node by a single minimalpath within a given time under a budget restriction. From the point ofquality management, it is the probability of satisfying transmissionrequirement in a specific time, which is treated as a performance indexof the service system.

For evaluating the system reliability of a real-life flow network, acomputer is used in the present invention to run a reliabilityevaluation software which provides a network model for simulating thereal-life flow network.

Refer to FIG. 2 for a block diagram of the hardware in the systemreliability evaluation method for routing policy according to thepresent invention. A computer 100 has an input unit 110, an operationunit 120, a storage unit 140 and an output unit 150. The input unit 110is a keyboard or a handwriting input device. The operation unit 120 is acentral processing unit (CPU). The storage unit 140 is a hardwareelectrically connected to the input unit 110, the operation unit 120 andthe output unit 150. A reliability evaluation software 130 is installedin the hardware. The output unit 150 is a display or a printer.

Refer to FIG. 3 for a flow chart of the system reliability software 140executing the system reliability evaluation method for routing policyaccording to the present invention. The method includes the steps of:

Step (S200): building a virtual network to correspond with the real-lifeflow network in the network model according to number of the nodes N andthe arcs a_(i) in the real-life flow network. Supposing that the networkis a binary-state system, and each arc has two cases of normal andfailure. All minimal paths P_(j)={a_(j1), a_(j2), . . . , a_(jn) _(j) }between the start node s to the terminal node t in the virtual networkare selected. The minimal path is required to be an ordered sequence ofthe arcs a_(i) between the start node s to the terminal node t withoutloops.

Step (S201): receiving a time restriction T and a budget restriction Bset by the user.

Step (S202): inputting the transmission requirement d of goods,commodities or data from the input unit 110 by user.

Step (S203): given the transmission requirement d and the timerestriction T, investigating flow distribution of the flow network underthe minimal path, distributing the transmission requirement d in a firstminimal path P₁ of the virtual network between the start node s and theterminal node t for calculating the flow in the first minimal path P₁under the time restriction T.

Step (S204): calculating a maximum flow of each arc a_(i) per unit timeaccordingly, which is regarded as a capacity x_(i) of the arc a_(i)according to the flow in the first minimal path P₁. Getting a capacityvector X≡(x₁, x₂, . . . , x_(n)) to represent the current state of eacharc a_(i) of the first minimal path P₁. The capacity vector X iscomposed by the capacities x₁, x₂, . . . , x_(n) of the arcs. Thecapacities are stochastic to correspond with the flow distribution ofthe flow network.

Under the capacity vector X of certain flow distribution, the operationunit checking if the transmission cost F(P_(j)) satisfies the budgetrestriction B for sending the transmission requirement d in the minimalpath. Defining a lower boundary vector of making the flow networksatisfy the lowest boundary of the time restriction T and the budgetrestriction B. Any capacity vector larger than the lower boundary vectorcan satisfy the requirement of sending the transmission requirement dunder the time restriction T and the budget restriction B. In otherwords, from the capacity vectors X relative to the flow distribution,all lower boundary vectors lower than or equal to the time restriction Tand the budget restriction B can be selected. Applyinginclusion-exclusion rule, disjoint-event method or state-spacedecomposition to calculate a first feasible probability Pr(S₁) of thecapacity vector X larger than or equal to the lower boundary vector,which is the probability that the flow network satisfies thetransmission requirement d.

Step (S205): judging if the state of every arc in the first minimal pathP₁ is feasible.

Step (S206): if one of the states of all arcs in the first minimal pathP₁ is unfeasible, calculating a first unfeasible probability Pr(E₁),which is the probability that the capacity vector X is unable to satisfythe lower boundary vector in the virtual network.

Step (S207): distributing the transmission requirement d in a secondminimal path P₂ of the virtual network between the start node s and theterminal node t for calculating the flow in the second minimal path P₂under the time restriction T as step (S203).

Step (S208): calculating a second feasible probability Pr(S₂) of thecapacity vector X larger than or equal to the lower boundary vector asstep (S204).

Step (S209): judging if the state of every arc in the second minimalpath P₂ is feasible, yes for performing step (S210), no for executingthe process from step (S206) to step (S208) for the next minimal path.

Step (S210): due to many possibilities of the lower boundary vectorcalculated in step (S204) and step (S208), integrating said firstfeasible probability, said first unfeasible probability and said secondfeasible probability into a system feasible probability, which is theprobability that the flow network satisfies the transmission requirementd, called the system reliability, represented by R_(d,T,B).

Additionally, an expectation of transmission ability of the flow networkis defined as product of the system reliability R_(d,T,B) and thetransmission requirement d, and an expectation of transmission time ofthe flow network is defined as product of the system reliabilityR_(d,T,B) and the time restriction T. Thus, after calculating the systemreliability R_(d,T,B),

$\sum\limits_{d}{R_{d,T,B} \times d}$

is the expected transmission ability of the flow network and

$\sum\limits_{T}{R_{d,T,B} \times T}$

is the expected transmission time of sending d units of data under thetime restriction T.

After calculating the system reliability, the specified routing policyis analyzed by said system reliability to find the arcs and the nodesinfluentially injuring or contributing to the transmitting efficiency ofthe flow network.

Refer to FIG. 1 for a benchmark network to illustrate the proposedalgorithm. The algorithm and an embodiment are presented in followingtext.

Let G≡(N, A, L, M, C) denotes a stochastic-flow network where N denotingthe set of nodes, A {a_(i)|1≦i≦n} denoting the set of arcs, L≡(l₁, l₂, .. . , l_(n)) with l_(i) denoting the lead time of a_(i) and M≡(M₁, M₂, .. . , M_(n)) with M_(i) denoting the maximal capacity of a_(i),C≡{c_(i)|1≦i≦n} with c_(i) denoting the transmission cost of arc a_(i)per unit of data. The capacity is the maximal number of data sentthrough the medium (an arc or a path) per unit of time. In thestochastic-flow network, the current capacity of arc a_(i) isstochastic, denoted by x_(i), taking values 0=b_(i1)<b_(i2)< . . .<b_(ir) _(i) =M_(i), where b_(ij) is an integer for j=1, 2, . . . ,r_(i). The vector X≡(x₁, x₂, . . . , x_(n)) denotes the capacity vector.

If the flow in the flow network is able to satisfy the transmissionrequirement d and the capacity of the arcs under the time restrictionand the budget restriction at the same time, the transmission of theflow network is defined as a success.

In this flow network, assuming each node N is perfectly reliable, thecapacities of different arcs are statistically independent and alltransmission requirement such as data and commodities are sent throughone minimal path. The comparisons of vectors are:

Y≧X (y₁, y₂, . . . , y_(n))≧(x₁, x₂, . . . , x_(n)): y_(i)≧x_(i) foreach i=1, 2, . . . , n.

Y≧X (y₁, y₂, . . . , y_(n))>(x₁, x₂, . . . , x_(n)): Y≧X and y_(i)>x_(i)for at least one i.

Suppose that P₁, P₂, . . . , P_(m) are minimal paths of G from start sto terminal t. With respect to each P_(j)={a_(j1), a_(j2), . . . ,a_(jn) _(j) }, the maximum capacity under the capacity vector

$X\mspace{14mu} {is}\mspace{14mu} {\min\limits_{1 \leq k \leq n_{j}}{\left( x_{jk} \right).}}$

If d units of data are required to be transmitted through minimal pathP_(j) under the capacity vector X and the budget restriction B, then thetransmission time, denoted by Ψ(d, X, B, P_(j)), is lower than or equalto the time restriction, represented by:

$\begin{matrix}{{{{{lead}\mspace{14mu} {time}\mspace{14mu} {of}\mspace{14mu} P_{j}} + \left\lceil \frac{d}{{the}\mspace{14mu} {capacity}\mspace{14mu} {of}\mspace{14mu} P_{j}} \right\rceil} = {{\sum\limits_{k = 1}^{n_{j}}l_{jk}} + \left\lceil \frac{d}{\min\limits_{1 \leq k \leq n_{j}}\; x_{jk}} \right\rceil}},} & (1)\end{matrix}$

where ┌x┐ is the smallest integer such that ┌x┐≧x. Any capacity vector Xwith Ψ(d, X, B, P_(j))≦T means that the network can send d units of datafrom the start node s to the terminal node t within time restriction Tunder the capacity vector X and budget restriction B.

If X is a minimal capacity vector such that the network can send d unitsof data within T units of time, then X is called a lower boundaryvector. It is trivial that (i) Ψ(d, X, B, P_(j))≦T and (2) Y<X, Ψ(d, X,B, P_(j))>T for any capacity vector. Ω_(j) represents the set of thecapacity vectors X and Ω_(j,min) represents the set of the lowerboundary vectors. The system reliability R_(d,T,B) is a feasibleprobability Pr{X|Ψ(d, X, B, P_(j))≦T} at this time.

Several methods such as inclusion-exclusion rule, disjoint-event methodand state-space decomposition are able to be applied to calculate thesystem reliability R_(d,T,B). Note that Pr{X≧Y}=Pr{x₁≧y₁}×Pr{x₂≧y₂}× . .. ×Pr{x_(n)≧y_(n)} if Y=(y₁, y₂, . . . , y_(n)).

The algorithm to evaluate the system reliability is as follows:

Step 1. For each minimal path P_(j)={a_(j1), a_(j2), . . . , a_(jn) _(j)}, find the minimal capacity vector Z_(j)=(z₁, z₂, . . . , z_(n)) suchthat the network sends d units of data within T units of time.

1. Find the minimal capacity h of the minimal path P_(j) such that dunits of data can be sent through the minimal path P_(j) within T unitsof time. That is, find the smallest integer h such that

$\begin{matrix}{{{\sum\limits_{k = 1}^{n_{j}\;}l_{jk}} + \left\lceil \frac{d}{h} \right\rceil} \leq {T.}} & (2)\end{matrix}$

2. If

$h \leq {\min\limits_{1 \leq k \leq n_{j}}\left( M_{jk} \right)}$

then Z_(j) can be obtained according to:

$\begin{matrix}\left\{ \begin{matrix}{z_{i} = {{{the}\mspace{14mu} {minimal}\mspace{14mu} {capacity}\mspace{14mu} u\mspace{14mu} {of}\mspace{14mu} a_{i}\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} u} \geq v_{j}}} & {\; {{{{if}\mspace{14mu} a_{i}} \in P_{j}},}} \\{{z_{i} = 0}\mspace{484mu}} & {{{if}\mspace{14mu} a_{i}} \notin {P_{j}.}}\end{matrix} \right. & (3)\end{matrix}$

Otherwise, Z_(j) does not exist.

Step 2. For each minimal path P_(j)={a_(j1), a_(j2), . . . , a_(jn) _(j)}, check if it satisfies the budget restriction B.

1. Calculate the transmission cost

${F\left( P_{j} \right)} = {\sum\limits_{i = 1}^{n_{j}}\left( {d \cdot c_{ji}} \right)}$

of the minimal path P_(j).

2. If F(P_(j))>B, then the lower boundary vector Z_(j) does not exist.

Step 3. If Z_(j) exists, then B_(j)={X|X≧Z_(j)}; Otherwise, B_(j)=φ.Then the system reliability is

$\Pr {\left\{ {\overset{m}{\bigcup\limits_{j = 1}}B_{j}} \right\}.}$

The network administrator decides the routing policy in advance toindicate a first priority minimal path (MP), a second priority minimalpath (MP) (or named alternative MP), etc.

Supposing that the first minimal path is the first priority minimalpath, the second minimal path, which is the alternative second priorityminimal path, will be responsible for the transmission duty if the firstpriority minimal path fails. The minimal path fails if and only if atleast one arc in it fails. Such that the level of routing policy iscalled a first level. Similarly, supposing that the second minimal pathis the second priority minimal path, the third minimal path, which is analternative third priority minimal path, will be responsible for thetransmission duty if the second priority minimal path fails. Such thatthe level of routing policy is a second level if two alternative MPs arestanding by. In this embodiment, perform said evaluation method forrouting policy in flow network to calculate the system reliability atthe first level. Let E_(j) denote the event that P_(j) fails, and S_(j)denote the event that P_(j) is able to send d units of data within timeT. Then, the unfeasible probability is:

$\begin{matrix}{{{\Pr \left( E_{j} \right)} = {{\Pr \left( {x_{i} = {{0\mspace{14mu} {for}\mspace{14mu} {at}\mspace{14mu} {least}\mspace{14mu} {one}\mspace{14mu} a_{i}} \in P_{j}}} \right)} = {1 - {\prod\limits_{i:{a_{i} \in P_{j}}}{\Pr \left( {x_{i} \geq 1} \right)}}}}},} & (4) \\{{j = 1},2} & \;\end{matrix}$

Indicate that Ω_(j,min)={Z_(j)} where Z_(j) is generated from thealgorithm. Thus, the jth feasible probability, denoted by Pr(S_(j)), isrepresented by:

$\begin{matrix}{{{\Pr \left( S_{j} \right)} = {{\Pr \left\{ \Omega_{j,\; {m\; i\; n}} \right\}} = {{\Pr \left\{ X \middle| {X \geq Z_{j}} \right\}} = {\prod\limits_{i:{a_{i} \in P_{j}}}{\Pr \left( {x_{i} \geq z_{i}} \right)}}}}},{j = 1},2.} & (5)\end{matrix}$

The system reliability R_(d,T,B) is the feasible probability that thenetwork is able to send d units of data within time T. If the state ofevery arc in the second minimal path is feasible, the system reliabilityR_(d,T,B) is

$\begin{matrix}\begin{matrix}{R_{{d,T,B}\;} = {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \middle| E_{1} \right)} \times {\Pr \left( E_{1} \right)}}}} \\{= {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \right)} \times {\Pr \left( E_{1} \right)}}}} \\{= {{\prod\limits_{i:{a_{i} \in P_{1}}}{\Pr \left( {x_{i} \geq z_{i}} \right)}} + {\prod\limits_{i:{a_{i} \in P_{2}}}{{\Pr \left( {x_{i} \geq z_{i}} \right)} \times {\left( {1 - {\prod\limits_{i:{a_{i} \in P_{1}}}{\Pr \left( {x_{i} \geq 1} \right)}}} \right).}}}}}\end{matrix} & (6)\end{matrix}$

Step 1 and step 2 are alternative with each other. Use the benchmarknetwork in FIG. 1 to illustrate the proposed algorithm. The capacity,budget, and the lead time of each of the arcs are shown in Table 1. In afirst embodiment, if 20 units of data are required to be sent from startto terminal within 15 units of time under 200 units of budget. Then alllower boundary vectors and the system reliability to meet such arequirement can be derived as follows:

TABLE 1 The arc data of FIG. 1 Arc Capacity Probability Lead time a1 5^(a) 0.85 2 3 0.05 1 0.05 0 0.05 a2 5 0.80 1 3 0.10 1 0.05 0 0.05 a3 40.85 3 2 0.05 1 0.05 0 0.05 a4 3 0.90 3 1 0.05 0 0.05 a5 5 0.85 2 3 0.051 0.05 0 0.05 a6 6 0.80 2 4 0.05 2 0.05 1 0.05 0 0.05 a7 4 0.85 3 2 0.051 0.05 0 0.05 a8 2 0.95 1 0 0.05 a9 5 0.60 2 3 0.20 2 0.10 1 0.05 0 0.05a10 7 0.65 2 5 0.10 4 0.10 2 0.05 1 0.05 0 0.05 a11 6 0.70 1 4 0.10 20.10 1 0.05 0 0.05 a12 2 0.95 2 0 0.05 ^(a)Pr{the capacity of a₁ is 5} =0.85.

For the first minimal path P₁={a₁, a₂, a₃}:

1. The transmission cost F(P₁)=20×(3+4+1)=160 is smaller the budgetrestriction B.

2. The lead time of P₁ is l₁+l₂+l₃=6. Then v₁=3 is the smallest integersuch that

$\left( {6 + \left\lceil \frac{20}{v_{1}} \right\rceil} \right) \leq 15.$

2. The maximal capacity of P₁ is min{5,5,4}=4. So, z₁=3, z₂=3, z₃=4 andz_(i)=0 for others. Thus, Z₁=(3,3,4,0,0,0,0,0,0,0,0,0).

For the second minimal path P₂={a₅, a₆, a_(n)}:

1. The transmission cost F(P₂)=20×(3+4+1)=160 is smaller the budgetrestriction B.

2. The lead time of P₂ is l₅+l₆+l₇=7. Then v₂=3 is the smallest integersuch that

$\left( {7 + \left\lceil \frac{20}{v_{2}} \right\rceil} \right) \leq 15.$

3. The maximal capacity of P₂ is min{5,6,4}=4. So, z₅=3, z₆=4, z₇=4 andz_(i)=0 for others. Thus, Z₂=(0,0,0,0,3,4,4,0,0,0,0,0).

In the process to evaluate the system reliability R_(20,15,200),

$\begin{matrix}\begin{matrix}{{\Pr \left( S_{1} \right)} = {\Pr \left\{ X \middle| {X \geq \left( {3,3,4,0,0,0,0,0,0,0,0} \right)} \right\}}} \\{= {\Pr \left\{ {x_{1} \geq 3} \right\} \times \Pr \left\{ {x_{2} \geq 3} \right\} \times \Pr \left\{ {x_{3} \geq 4} \right\} \times \Pr \left\{ {x_{4} \geq 0} \right\} \times}} \\{{\Pr \left\{ {x_{5} \geq 0} \right\} \times \Pr \left\{ {x_{6} \geq 0} \right\} \times \Pr \left\{ {x_{7} \geq 0} \right\} \times \Pr \left\{ {x_{8} \geq 0} \right\} \times \Pr \left\{ {x_{9} \geq 0} \right\} \times}} \\{{\Pr \left\{ {x_{10} \geq 0} \right\} \times \Pr \left\{ {x_{11} \geq 0} \right\} \times \Pr \left\{ {x_{12} \geq 0} \right\}}} \\{= {0.9 \times 0.9 \times 0.85}} \\{{= 0.6885},}\end{matrix} & \; \\\begin{matrix}{{\Pr \left( S_{2} \right)} = {\Pr \left\{ X \middle| {X \geq \left( {0,0,0,0,3,4,4,0,0,0,0,0} \right)} \right\}}} \\{= {\Pr \left\{ {x_{5} \geq 3} \right\} \times \Pr \left\{ {x_{6} \geq 4} \right\} \times \Pr \left\{ {x_{7} \geq 4} \right\}}} \\{= {0.9 \times 0.85 \times 0.85}} \\{= {0.65025\;.}}\end{matrix} & \; \\\begin{matrix}{{\Pr \left( E_{1} \right)} = {1 - {\Pr \left\{ {x_{1} \geq 1} \right\} \times \Pr \left\{ {x_{2} \geq 1} \right\} \times \Pr \left\{ {x_{3} \geq 1} \right\}}}} \\{= {0.1426525\;.}}\end{matrix} & \; \\\begin{matrix}{R_{20,15} = {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \right)} \times {\Pr \left( E_{1} \right)}}}} \\{= {0.6885 + {0.65025 \times 0.142625}}} \\{= {0.781241906\;.}}\end{matrix} & \;\end{matrix}$

In a second embodiment, the unit of the time restriction T is loosenedto be 18 from 15, then v₁=2, v₂=2, Z₁=(3,3,2,0,0,0,0,0,0,0,0,0) andZ₂=(0,0,0,0,3,2,2,0,0,0,0,0). The system reliability R_(20,18,200)increases to 0.832973625 where Pr(S₁)=0.729 and Pr(S₂)=0.729.

In FIG. 1, P₁={a₁, a₂, a₃}, P₂={a₅, a₆, a₇}, and P₃={a₉, a₁₀, a₁₁} arethree disjoint minimal paths (MPs) in the virtual network. Under therequirement that d=20, B=200 and T=15, it is known that Pr(S₁)=0.6885and Pr(S₂)=0.65025 from the first embodiment. Obtain a third feasibleprobability Pr(S₃)=0.729 after running the algorithm for a third minimalpath P₃. In a third embodiment, the third minimal path P₃ is the firstpriority minimal path and the first minimal path P₁ is the secondpriority minimal path according to the step. 2 and the step. 3 form thealgorithm. Under this routing policy,R_(20,15,200)=Pr(S₃)+Pr(S₁)×Pr(E₃)=0.729+0.6885×0.142625=0.829693125which is larger than the system reliability with respect to the routingpolicy in the first embodiment. Table 2, which lists the systemreliabilities with respect to different routing policies, shows theoptimal routing policy is obtained from the proposed procedure.

TABLE 2 System reliabilities for different routing policies Firstpriority MP P₁ P₂ P₁ P₃ P₂ P₃ Second priority P₂ P₁ P₃ P₁ P₃ P₂ MPSystem reliability 0.7812419 0.7484473 0.7924736 0.8296931 0.75422360.8217419

In an embodiment, for the routing policy with the second level, thesecond priority minimal path takes charge of the transmission duty ifthe first priority MP fails, and the third priority MP takes charge ifthe second priority MP fails. Under the routing policy with the secondlevel, the system reliability R_(d,T,B) is

$\begin{matrix}\begin{matrix}{R_{d,T,B} = {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \middle| E_{1} \right)} \times {\Pr \left( E_{1} \right)}} + {{\Pr \left( S_{3} \middle| {E_{1}E_{2}} \right)} \times {\Pr \left( {E_{1}E_{2}} \right)}}}} \\{= {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \right)} \times {\Pr \left( E_{1} \right)}} + {{\Pr \left( S_{3} \right)} \times {\Pr \left( E_{1} \right)} \times {\Pr \left( E_{2} \right)}}}} \\{= {{\prod\limits_{i:{a_{i} \in P_{1}}}{\Pr \left( {x_{i} \geq z_{i}} \right)}} + {\prod\limits_{i:{a_{i} \in P_{2}}}{{\Pr \left( {x_{i} \geq z_{i}} \right)} \times \left( {1 - {\prod\limits_{i:{a_{i} \in P_{1}}}{\Pr \left( {x_{i} \geq 1} \right)}}} \right)}} +}} \\{{\prod\limits_{i:{a_{i} \in P_{3}}}{{\Pr \left( {x_{i} \geq z_{i}} \right)} \times \left( {1 - {\prod\limits_{i:{a_{i} \in P_{1}}}{\Pr \left( {x_{i} \geq 1} \right)}}} \right) \times}}} \\{{\left( {1 - {\prod\limits_{i:{a_{i} \in P_{2}}}{\Pr \left( {x_{i} \geq 1} \right)}}} \right).}}\end{matrix} & (7)\end{matrix}$

Utilize the data in the first embodiment to evaluate the systemreliability if P₁, P₂ and P₃ are the first, second and third priorityMP, respectively. Then,

$\begin{matrix}{R_{20,15,200} = {{\Pr \left( S_{1} \right)} + {{\Pr \left( S_{2} \right)} \times {\Pr \left( E_{1} \right)}} + {{\Pr \left( S_{3} \right)} \times {\Pr \left( E_{1} \right)} \times {\Pr \left( E_{2} \right)}}}} \\{= {0.6885 + {0.65025 \times 0.142625} + {0.729 \times 0.142625 \times 0.142625}}} \\{= {0.796071144.}}\end{matrix}$

In another embodiment, as the sort criteria in the algorithm todetermine the optimal routing policy, P₃, P₁ and P₂ should be orderedfrom the first, second and third priority MP, respectively. Then, thesystem reliability increases to 0.840424626 as follows.

$\begin{matrix}{R_{20,15,200} = {{\Pr \left( S_{3} \right)} + {{\Pr \left( S_{1} \right)} \times {\Pr \left( E_{3} \right)}} + {{\Pr \left( S_{2} \right)} \times {\Pr \left( E_{3} \right)} \times {\Pr \left( E_{1} \right)}}}} \\{= {0.729 + {0.6885 \times 0.142625} + {0.65025 \times 0.1426525 \times}}} \\{0.142625} \\{= {0.840424626\;.}}\end{matrix}$

In order to obtain higher system reliability, extend the above procedureto the routing policy with a third level, a fourth level, and so on.

Above all, users need to input the transmission requirement d, thebudget restriction B, the time restriction T and the lead time Laccording to the present invention. The lead time L depends on theprocessing time of the given amount of data in real-life flow network,such as computer system, telecommunication system, logistic system andtransportation system. Based on the inputted data, the systemreliability can be outputted in terms of flow distribution, budget checkand time check.

Actually, the present method is suitable for the system with time andcapacity characters, such as computer system, telecommunication systemand transportation system. From the point of quality management, thesystem reliability can be regarded as a performance index. The presentmethod can be extended to a constrained quickest path problem, kquickest paths problem and all-pairs quickest path problem.

The foregoing description of the preferred embodiment of the inventionhas been presented for purposes of illustration and description. It isnot intended to be exhaustive or to limit the invention to the preciseform or to exemplary embodiments disclosed. Accordingly, the foregoingdescription should be regarded as illustrative rather than restrictive.Obviously, many modifications and variations will be apparent topractitioners skilled in this art. The embodiments are chosen anddescribed in order to best explain the principles of the invention andits best mode practical application, thereby to enable persons skilledin the art to understand the invention for various embodiments and withvarious modifications as are suited to the particular use orimplementation contemplated. It is intended that the scope of theinvention be defined by the claims appended hereto and their equivalentsin which all terms are meant in their broadest reasonable sense unlessotherwise indicated. Therefore, the term “the invention”, “the presentinvention” or the like is not necessary limited the claim scope to aspecific embodiment, and the reference to particularly preferredexemplary embodiments of the invention does not imply a limitation onthe invention, and no such limitation is to be inferred. The inventionis limited only by the spirit and scope of the appended claims. Theabstract of the disclosure is provided to comply with the rulesrequiring an abstract, which will allow a searcher to quickly ascertainthe subject matter of the technical disclosure of any patent issued fromthis disclosure. It is submitted with the understanding that it will notbe used to interpret or limit the scope or meaning of the claims. Anyadvantages and benefits described may not apply to all embodiments ofthe invention. It should be appreciated that variations may be made inthe embodiments described by persons skilled in the art withoutdeparting from the scope of the present invention as defined by thefollowing claims. Moreover, no element and component in the presentdisclosure is intended to be dedicated to the public regardless ofwhether the element or component is explicitly recited in the followingclaims.

1. A system reliability evaluation method for routing policy, using acomputer having an input unit, an operation unit and an output unit toexecute a reliability evaluation software which provides a virtualnetwork for simulating a flow network, the virtual network having astart node, a terminal node and plural arcs between the start node andthe terminal node for constituting plural minimal paths, the steps ofthe method comprising: inputting a transmission requirement, a timerestriction and a budget restriction from the input unit by users;distributing the transmission requirement in a first minimal path of thevirtual network; according to the transmission requirement, the timerestriction and the budget restriction, calculating a first feasibleprobability of the first minimal path in the virtual network; if thefirst minimal path is in an unfeasible state, distributing thetransmission requirement in a second minimal path of the virtualnetwork, and calculating a first unfeasible probability of the firstminimal path in the virtual network; according to the transmissionrequirement, the time restriction and the budget restriction,calculating a second feasible probability of the second minimal path inthe virtual network; integrating the first feasible probability, thefirst unfeasible probability and the second feasible probability into asystem feasible probability of the virtue network via the operationunit, and defining the system feasible probability as a systemreliability of the flow network; and displaying the system reliabilityon the output unit.
 2. The system reliability evaluation method forrouting policy of claim 1, further comprising: if the first minimal pathis in a feasible state, defining the first feasible probability as thesystem reliability.
 3. The system reliability evaluation method forrouting policy of claim 1, wherein the steps of distributing thetransmission requirement in one of the minimal paths comprise: selectingthe minimal paths of the virtual network, wherein each of the minimalpaths is an ordered sequence of the arcs between the start node to theterminal node without loops; calculating a flow in each of the minimalpaths; and transferring the flow in each of the minimal paths into acapacity of each of the arcs.
 4. The system reliability evaluationmethod for routing policy of claim 3, wherein the steps of calculatingthe flow of the minimal path comprise: providing a lead time for each ofthe arcs in the minimal path; with a transmission time lower than orequal to the time restriction, calculating the flow of the minimal path,wherein the transmission time equals to the sum of the lead time and thetransmission requirement divided by the flow of the minimal path; andjudging if a lower boundary vector exists, when the flow of the minimalpath is smaller than a maximum capacity of the minimal path.
 5. Thesystem reliability evaluation method for routing policy of claim 1,wherein the steps of calculating the feasible probabilities comprise:according to the transmission requirement, the time restriction, thebudget restriction and a transmission time which is smaller than orequal to the time restriction, calculating a maximum flow in each of thearcs in the minimal path per unit time, wherein the maximum flow is acapacity of the arc; defining a capacity vector comprising thecapacities of the arcs, the capacities being stochastic to correspondwith the flow distribution state of the flow network; executing a budgetcheck by the operation unit to check if a transmission cost satisfiesthe budget restriction for sending the transmission requirement in theminimal path; defining the capacity vector as a lower boundary vectorwhen the transmission cost and the transmission time of the minimal pathare less than or equal to the budget restriction and the timerestriction respectively; and calculating the probability that thecapacity vector of any path is larger than or equal to the lowerboundary vector of the minimal path, and defining said probability asthe feasible probability of the flow network.
 6. The system reliabilityevaluation method for routing policy of claim 5, wherein the steps ofthe budget check comprise: calculating the transmission cost of sendingthe transmission requirement in the minimal path; comparing the valuesof the transmission cost and the budget restriction; and according tothe comparison result, judging if the lower boundary vector of theminimal path exists.
 7. The system reliability evaluation method forrouting policy of claim 1, wherein a inclusion-exclusion rule, adisjoint-event method and a state-space decomposition are applied instep of calculating the feasible probabilities.
 8. The systemreliability evaluation method for routing policy of claim 1, furthercomprising: defining an expectation of the transmission ability of theflow network as product of the system reliability and the transmissionrequirement.
 9. The system reliability evaluation method for routingpolicy of claim 1, further comprising: defining an expectation of thetransmission time of the flow network as product of the systemreliability and the time restriction.